Your husband needs to walk and feed the dog right before he goes out the door. Then you do the after breakfast walking. Do you have the dog in a crate at night? This becomes an important "den" for the animal, a safe haven they know is their space. Puppies will cry at night because they miss their mother, but they still should sleep in a crate, and probably should either be in a crate or cordonned off in one room of the house while everyone is gone. Do you have newspaper down near the door (or in another spot) for those puppy emergencies?
Unfortunately, you gave in to the decision to get the dog when your kids are really too young to handle the responsibility of caring for it. Maybe your 5 year old can handle the feeding, you could start this on the weekends when you or your husband can provide a lot of oversight. I think it sends the wrong message to the kids if you bail out of the commitment just because it's difficult.
HOWEVER, if you see no light at the end of the tunnel, it is better to sell the dog than to let it wreck your lives, just make sure you and your husband make the decision together and that he understands how much more stressful this has made your life. Also, if you do give this dog away, don't get a new one as soon as you move into the new bigger house. Wait until your girls are old enough to handle the entire responsibility of care, the oldest should be at least 8 (but remember, on those rainy, miserable nights, and always the last walk before bed, MOM is the one who ends up with doggie duty, same goes for in house poo and vomit clean up, etc.). I would have the girls volunteer at a pet shelter and see if they still really want an animal when they learn more about the care involved.
Remember an adult dog is like a permanent two year old, but a puppy is a baby, make sure your family is ready for this responsibility before you do it again. Dogs can bring an immense amount of joy, but they always bring an immense amount of work. As for lying to your children, you're going to have to be really creative, and expect to hear from both of them how much they miss the dog (even years later from the 2 year old). Also, your 5 yo may not believe the lie, no matter how creative it is, and this could lead to trust issues.
Good luck, I hope you find a way to adjust to and love and care for the dog. But, if you really took on more than you can handle, I think the right thing is to get out, and don't feel guilty about it, just don't make the decision lightly, and again don't assume the bigger house will be the answer and dive right back in.